When you plug that into the LHS, the operator will annihilate the exponential, so there's no way you'll get the RHS out of that. Because the root m = 1 has multiplicity 3, your particular solution will have to look something like slapmaxwell1's candidate. I might recommend have something more like

That might be overkill, but it might not. A hunch: any of those constant that are superfluous will be easy to solve for, I think. So I don't think you'll waste too much time doing it this way.